Heat 1 1/2 tablespoons of the olive oil in a large skillet over medium-high heat until shimmering. Add the diced potatoes in a single layer and season them with the smoked paprika, garlic powder, onion powder, salt, and pepper. Stir once to coat, then let them sit undisturbed for 4 to 5 minutes until the bottoms are golden and crispy. Flip and cook another 4 to 5 minutes. The potatoes are ready when they are golden on the outside and just tender when poked with a fork, about 12 to 14 minutes total.

Push the potatoes to the edges of the skillet. Add the remaining 1/2 tablespoon of oil to the center and add the diced onion and bell pepper. Cook for 3 to 4 minutes, stirring occasionally, until the onion is softened and starting to turn translucent. Add the minced garlic and stir everything together for 30 seconds until fragrant.

Add the sliced turkey sausage to the skillet and stir to combine everything. If using raw sausage, cook for 5 to 6 minutes, stirring every couple of minutes, until the sausage is browned and cooked through with no pink. If using fully cooked sausage, cook for 3 to 4 minutes just to get some color on the slices. Taste and adjust salt and pepper as needed.

If you are adding fried eggs, push the hash to the sides of the skillet to clear four small spaces, or fry the eggs in a separate nonstick pan. Crack an egg into each space, cover the skillet loosely with a lid or a piece of foil, and cook over medium heat for 2 to 3 minutes until the whites are set but the yolks are still runny. Cook 1 minute longer if you prefer firm yolks.

Spoon the hash into bowls or onto plates and top each serving with a fried egg if using. Scatter chopped fresh parsley over the top and serve immediately while the potatoes are still crispy.